ICALP 1994: Automata, Languages and Programming pp 520-531

# The size of an intertwine

• Jens Lagergren
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 820)

## Abstract

An intertwine of two graphs H and H′ is a graph G such that G contains both H and H′ as minors, but no proper minor of G contains both H and H′ as minors. We give an upper bound on the size of an intertwine of two given planar graphs. For two planar graphs H and H′, the bound is triply exponential in O(m5) where m≤max(¦V(H)¦, ¦V(H′)¦). We also give an upper on the size of an intertwine of two given trees T and T′. This bound is exponential in O(m3 log m) where m≤max(¦V(T)¦, ¦V(T′)¦). Let O1 be the set of obstructions for a minor closed family L1 and O2 the set of obstructions for a minor closed family L2. It is a natural to ask the following question: how do we given O1 and O2 compute the obstructions for L1L2 and L1L2. Both these sets of obstructions are known to be finite, and to obtain the obstructions for L1L2 is easy. However, to compute the obstructions for L1∪ L2 is hard. Our upper bound enables us to given O1 and O2 compute a bound on the size of any obstruction for L1L2 whenever L1 and L2 are families of planar graphs.

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